Estimation of local scour depth around twin piers using gene expression programming (local scour around twin piers) Open Access

d

Pier diameter

D₅₀

Mean particle size of the sediment

H

Flow depth above the sand bed

ρ

Water density

ρ_s

Density of sediments

C_s

Center-to-center pier spacing

Q

Volumetric rate of flow

t

Time taken for each experimental evaluation

t_e

Time taken for achieving equilibrium scour depth

V

Velocity of flow

V_c

Critical velocity of flow

Scour depth

Y_su

Scour depth upstream for the single pier

Y_sul

Scour depth on the upstream of the left pier

Y_sur

Scour depth on the upstream of the right pier

Percentage of scour depth attained at equilibrium time

Bridge scour is induced by the erosive action of streaming water and the transport of sediments around piers and abutments of bridges. Scouring around the piers significantly reduces the amount of support provided to the foundations of hydraulic structures, and may lead to massive bridge collapses. These collapses are responsible for considerable expense in maintenance, replacement, and the possibility of environmental damage (Coleman & Melville 2001). As per the Federal Highway Administration (FHWA) report (Arneson et al. 2012), in the USA, 60% of the bridge failures in the last two decades have been due to hydraulic failures. According to research conducted previously (Afzali 2015), around 1,000 of the 600,000 bridges in the United States have collapsed due to scouring, accounting for nearly half of the collapses. Furthermore, research reported in one study (Link et al. 2020) states that 50% of the bridge failures across the world are due to scouring. Therefore, it becomes essential for researchers to accurately predict scour depths and advert these effects. Due to the complex interaction between unsteady flow patterns, erodible sediments, and submerged structures, the accurate dependency of these parameters on scour depth remains an unsolved challenge (Liang et al. 2019).

To reduce the risk of bridge failure, engineers need to use accurate scour predictions and effective countermeasures. Many mathematical models have been suggested during the last few decades based on experimental and theoretical studies, but few of them are discussed here. The work conducted by Sheppard et al. (2014) compiled research and field data to evaluate 23 of the most recent and widely used equilibrium local scour equations for cohesionless sediments. Based on the derived results, six equations were chosen for the final assessment. These are Jain (Jain & Fischer 1979), Froehlich (Froehlich 1988), Melville (Melville 1997), Hydraulic Engineering Circular No. 18 (HEC-18) (Richardson & Davis 2001), modified HEC-18 (Arneson et al. 2012), and Sheppard and Melville (Sheppard et al. 2014). Similarly, the authors (Hamidifar et al. 2021) examined the 10 most used scour depth estimation equations combined with eight critical velocity equations. Results of the study suggested three hybrid models (a) Jain and Fisher (Jain & Fischer 1979) and Richardson and Davis (Richardson & Davis 2001), (b) Jain (Jain 1981), and Arneson (Arneson et al. 2012) and (c) Jain and Fisher (Jain & Fischer 1979), which outperformed the previously reported scour depth results. In the latest work (Vijayasree & Eldho 2021) the author proposed a modification in terms of semi-empirical equation to follow the Indian road congress equation. These investigations relied on a single-pier study and empirical models based on standard statistical regression techniques, which could not identify the enormously complicated and non-linear relationship between scour depth and its primary causes. However, the majority of these equations that are considered to be trustworthy were established some decades ago. As a result, these may not be able to precisely predict scour for modern long-span bridges with large piers or multiple foundation types (Liang et al. 2019).

Flow field and local scour around circular bridge piers have been extensively researched computationally and experimentally during the last few decades. Latest and significant research in this category include several previous reports (Khan et al. 2017; Qi et al. 2019; Nagel et al. 2020; Pandey et al. 2020a; Bordbar et al. 2021; Jain et al. 2021; Malik et al. 2021). Furthermore, several investigations on a set of piers have been also conducted by researchers, such as the work of Sumner et al. (1999) who used flow visualisation, particle image velocimetry (PIV) and hot-film anemometry to explore the flow field for two and three circular cylinders positioned side by side, with the centre-to-centre pitch ratio expanding. Their observations reinforced the concept that the arrangement of circular cylinders needs Reynolds number independence. Similarly, the work of Akilli et al. (2004) utilised the PIV approach to analyse flow around two and three side by side cylindrical piers, and findings indicated that in the case of side-by-side bridge piers, the flow structure behind the cylinders is asymmetrical and that in the case of three cylinders both an asymmetrical and a symmetrical flow were observed. In addition, authors (Ataie-Ashtiani & Aslani-Kordkandi 2012) conducted an experimental study to analyse the flow characteristics of two piers placed in a side-by-side configuration and validated the results with the numerical simulations. It is concluded that the impacts of these flow vortices between the piers should be incorporated into the semi-empirical equations for a better estimation of the scour depth for side-by-side piers arrangement. All of the above-mentioned literature studies were focused on the flow field around the two and three piers. Conversely, only a few studies are available in the literature that provide regression-based equations for the group of piers Coleman et al. (2005; Gaudio et al. 2013; Amini Baghbadorani et al. 2018). The work in Coleman et al. (2005) proposed a new methodology named the Melville and Coleman equation to evaluate local scour depth at a complex pier. HEC-18 and Florida Department of Transportation (FDOT) equations are reviewed in Amini Baghbadorani et al. (2018), and a new equation for the protective benefit of pile cap frontal extension was suggested. Despite extensive laboratory studies, the regression-based equations have involved tedious calculations and have not provided promising predictions (Gaudio et al. 2013).

In very recent work, researchers utilised an artificial intelligence (AI)-based approach to predict scour depth (Sharafati et al. 2020). Soft computing approaches have simplified this task and made it more acceptable and trustworthy than traditional ways of analysis (Muzzammil et al. 2015). artificial neural networks (ANN), genetic programming (GP), genetic algorithms (GA), group method of data handling (GMDH), gene expression programming (GEP), adaptive network-based fuzzy inference system (ANFIS), and radial basis function (RBF) are among the AI techniques now being used to solve various hydraulics engineering problems. Some studies (Akib et al. 2014; Choi et al. 2017) utilised the ANFIS model for solving different scour problems; others (Pandey et al. 2020b, 2020c) utilised a GA approach for scour depth prediction. Furthermore, other work (Najafzadeh & Azamathulla 2013; Najafzadeh et al. 2015) presented an application of GMDH to predict the scour depth around bridge pier. It utilised the Levenberg–Marquardt (LM) method and compared this finding with ANFIS, RBF-NN, and certain empirical equations. It was determined that the GMDH-LM method offered more reliable predictions than the others. Even while ANN models outperform classic regression-based methods, they cannot give a direct relationship between scour depths and the variables that influence them, unlike GEP. In addition, other studies (Guven & Gunal 2008; Azamathulla et al. 2012; Muzzammil et al. 2015; Najafzadeh et al. 2016; Bateni et al. 2019) have used GEP to evaluate scour depth and compared the findings with other regression-based equations, concluding that GEP is the most estimable modelling methodology for scour depth evaluation, among other tools.

In general, computational assessment of scour depth has been undertaken based on AI methodologies and GEP in particular. This methodology is not frequently used, and there is an imperative need to carry out research in this area. Hence, the present study evaluates the capability of the GEP for predicting the scour depth around the twin piers configured side by side. In addition, to corroborate our proposed results, an experimental study was carried out which depicted the characteristics of flow around the piers and to train the proposed formulation. Finally, the precision ability of the proposed GEP formulation was compared with the experimental data, as well as regression-based equations that are previously reported in the literature.

This paper addresses the problem of local scour estimation encircling twin bridge piers that are aligned in the transverse direction to the flow, and a solution is provided by utilising the GEP modelling technique and is validated experimentally. The contributions of this study are as follows:

• Experimental investigation to assess the influence of flow rate and centre-to-centre spacing on local scour around the piers.

• GEP technique is used for estimating scour depth.

• Sensitivity analysis for assessing the influence of various parameters.

• The prediction capability of GEP is compared with the existing semi-empirical equations.

To the best of the authors' knowledge, this study is the first to perform the scour depth estimation using GEP for twin piers positioned in a side-by-side manner and validated with experimental results.

Experimental procedures are carried out at the Hydraulic Engineering Laboratory, Delhi Technological University, India. Experiments were conducted in 14 m long, 1.10 m wide, and 0.80 m deep recirculating flume equipped with an acoustic Doppler velocimeter (ADV) and a Venturi-meter. The schematic diagram of the experimental set-up is depicted in Figure 1 and laboratory set-up in Figure 2. The working section of 5 m length, 1.10 m width, and 0.20 m depth in the form of the sand recess was located 5 m downstream from the inlet section of the flume. The upstream section of the working area was filled with gravel to develop the flow, and the rest of the section was filled with the sand of specific gravity (S_g) of 2.62 and mean particle size (D₅₀) of 0.60 mm. The mean particle size of the sediment was chosen carefully to negate the effect of sediment on the scour development pattern. As stated by Raudkivi & Ettema (1983) and Melville & Chiew (1999) that if D/D₅₀>50, then the effect of sediment is present and, this ratio is 83.33 for the present study, so this effect can be neglected. The value of uniformity coefficient, C_u, is 2, and curvature coefficient, C_c, is 1, and if C_u <6 and 1< C_c <3 for sand, then it will be taken into consideration as uniformly graded (Arneson et al. 2012). Flow depth in the working section is regulated using a sluice gate mounted downstream of the working section. The experimental investigation is categorised into two configurations: (a) single pier, and (b) twin piers, placed transverse to the direction of flow. The rate of flow (Q) ranged from 0.0295 to 0.0537 m³/sec, water flow depth 0.12 to 0.163 m and centre-to-centre pier spacing (C_s) from 0 to 25 cm and other flow parameters ranges are listed in Table 1.

As depicted in Figure 3, experiments were carried out for eight centre-to-centre spacing (C_s) between the piers that are 1.5, 2, 2.5, 3, 3.5, 4, 4.5 and 5 times the diameter of the pier. Two cylindrical piers of acrylic pipes with a 5 cm diameter (D) were placed in the middle of the working section. The pier diameter was chosen carefully to have a minimum contraction effect; this effect is present if B/D < 3 and absent for B/D ≥ 5. The B/D ratio in the present study is greater than 5, and hence the boundary conditions recommended by Melville & Coleman (2000) are satisfied, and the contraction effect is ignored. The flow depth is kept in the range 0.12–0.163 m. The experimental duration for each experiment was set to 480 minutes as the scour depth generated after 300 minutes is deemed the equilibrium scour depth in accordance with the previously published work (Yanmaz & Altinbilek 1991; Mia & Nago 2003; Setia 2008). For continuously monitoring the pattern of scour evolution around both the piers, a vernier point gauge of precision ±0.1 mm is utilised. Vernier readings were recorded for 480 minutes in regular intervals, after every 30 minutes. After the completion of each experimental run, water from the flume section was drained out and scour depth around the twin piers was measured.

In this section, experimental results and discussion about scour characteristics are provided.

As stated earlier, experimental work is conducted in two configurations: (a) single pier and (b) twin pier. Figure 4 depicts the scour hole pattern observed after water drainage from the flume section for all the pier positioning. For the C_s = 1.5D one scour hole is formed around both the piers, and scoured sediment is deposited in the mid of both the piers in the form of one tail scour. For C_s = 2D a minor separation in the scour hole is observed, and only one tail scour deposition is noticed. For C_s = 2.5D, the formation of two separate scour holes started, but the tail scour one, and the outer circle is also one up to this pier spacing. At C_s = 3D to 4.5D, two separate scour holes are observed and scoured sediments are deposited in the mid of the two piers or along the centre line of flow. Up to C_s = 4.5D strong vortices (Horseshoe and wake) are formed between the piers and as a result a sediment deposition mound is formed at the middle of the two piers. Whereas in the case of C_s = 5D two separate holes are observed, but the deposition of sediment is observed at the rear of both the piers, due to the weak interference force and separate vortices formation around both the piers.

Results of the experimental study for both configurations are presented in Table 4. This includes observed values of MSD at the left and right pier. MSD values for both the piers are normalised with the scour depth at the single pier, Y_sul/Y_u and Y_sur/Y_u. From the table, it can be observed that the MSD increases with increase in the value of C_s/D, and attains its peak value at C_s/D = 2.5, then it starts decreasing up to C_s/D = 5. The MSD is observed at the upstream end of the pier for C_s/D = 2.5 and Q = 0.0537 m³/sec. It is represented in the pictorial and contour form in Figure 5(a) and 5(b), respectively. The analysis of the pier spacing effect is conducted on the basis of the scour hole developed after the lapse of 480 minutes. The scour holes formed around both piers merged, and the shape of the scour hole is identical to that of the single pier.

The scour holes formed around both piers merge in the C_s/D = 1.5 and 2 cases, and the shape of the scour hole is identical to that in the single-cylinder example. Even though the scour holes have merged in the C_s/D = 2.5 cases, there is some separation of the scour holes upstream of the cylinders. The scour depth between the two piers reduces as the distance between them expands because the jet-like flow, which drives flow acceleration and enhanced turbulence, is weakened. When the pier spacing C_s/D is more significant than 2.5, the local scour holes develop independently, resulting in isolated sedimentation mounds behind the individual piers. The pattern of the scour hole development for the 1.5 ≤ C_s/D ≤ 2.5 shows a highly complex behaviour due to the formation of large vortices between the piers.

Figure 6 illustrates a bar plot for the equilibrium scour depth for the single pier for seven rate of flow. It can be observed that the rate of flow and scour depth have a linear relationship with each other, that is, as the rate of flow increases, the scour depth around the pier also increases. The increasing pattern of scour development is the same for all the seven flow conditions; however, MSD depth continues to increase as the flow rate increases, as depicted in the plot.

Figure 7 depicts the equilibrium scour depth variation with seven different rates of flow in m³/sec. Scour trend for both the piers, the left pier and the right pier, is quite similar, as depicted in Figure 7(a) and 7(b), respectively. For the pier spacing ranges from 1.5 ≤ C_s/D ≤ 2.5, both piers started behaving as a single body with the radius double in size; as a result the scour hole that developed was rather larger in comparison with the single pier. For the pier spacing ranges from 2.5 < C_s/D < 4.5, the scour hole evolution is divided into two parts, however, the influence of the presence of the two piers in the close proximity is still evident. For the pier spacing C_s/D = 5, both the piers start behaving like individual piers with the development of two separate scour holes, however, the development of scour is greater than the single pier. From the results of this experimental study, it can be concluded that the maximum scour depth is achieved when the pier is placed in close proximity as observed for maximum scour depth for C_s/D for 1.5D and 2.5 D and the minimum scour depth was attained at the maximum spacing, that is 5D, due to the individual behaviour of the piers.

In one study (Ferreira 2006), Ferreira created GEP, which is an extension of GP. It combines the advantages of its forerunners: the GA and GP. GEP is a proficient computing technique of genotype/phenotype system, with the separate functioning of genotype and phenotype, whereas these functions are mixed in GP (Ferreira 2006; Guven & Gunal 2008). Due to the potential of GEP in terms of simple modelling, easy coding, and faster computations, it has gained popularity over other tools. GEP develops computer programs, which are subsequently encoded in linear chromosomes and generated or interpreted into expression trees (ETs). ETs are complex computer programs that, in most cases, are created to address a specific problem and are chosen based on their functionality. As genetic operators act at the chromosomal level, an advantage of the GEP technique is that it simplifies the creation of genetic variation. Genetic operations including mutation, inversion, transposition, and recombination are used to reconfigure chromosomes containing one or more genes. After that, one of the fitness function equation available in the literature is used to evaluate the fitness of each chromosome in the original population. The second advantage of GEP is that it is mutagenic, allowing for the growth of more sophisticated applications. Seeing these benefits, GEP seems an excellent choice for a perfectly mathematical model the scour depth. For more insight on GEP, readers are encouraged to see Ferreira (2006) and Teodorescu & Sherwood (2008).

The third step is to choose the architecture of the chromosomes, that is, head size and gene number. In the presented study, head size and gene number are chosen as ten and three, respectively. The fourth step is to select the connection function; in the presented study, a plus sign is chosen. The last step is to choose the genetic operator and its rate. In the present study, a genetic operator such as mutation and inversion in combination are utilised with the rate of three types of a transportation insertion sequence, root insertion sequence transposition, and gene transpiration.

Here, GEP formulation for the reliable forecasting of scour depth is provided. The variables and parameters that are used for formulating non-linear equations are enumerated in Tables 1 and 2, respectively. To formulate the GEP model, 112 testing data from the experimental investigation are utilised. Out of these 112 data sets, 67 (60% of the total) are chosen randomly as training data sets, and the remaining 45 (40% of the total) are chosen as testing data sets. The results of the GEP modelling are represented in Figure 8 in the form of ETs. Figure 8 consists of three ETs which are linked with each other by the plus sign. These gene pairs are a combination of constants whose values are also shown in the same Figure 8.

Figure 9 depicts a scatter-line plot of experimental observations versus GEP modelling predictions for relative scour depth for training and testing. It can be observed that the GEP results are the least dispersed from the experimental observations. The performance of the GEP modelling are analyses on the basis of a fitness function. The values of fitness function for training and testing are 0.00128 and 0.001424, respectively, while the correlation coefficients are 0.950 and 0.955, respectively. Consequently, GEP has proved its efficiency in generalizing the proposed GEP formulation and has effectively captured the non-linear relationships between the parameters.

The current study uses performance measures, R squared correlation (R²), RMSE and mean absolute error (MAE), to evaluate the performance of the various scour depth prediction equations. The definitions of these parameters are provided in the appendix. Table 3 shows the efficacy of the proposed GEP-based modelling in predicting the scour depth in comparison to previously reported results. It can be observed that it provided lower values of R², RMSE, and MAE. Overall, the findings of the presented study suggest that GEP-based modelling is a potential alternative to existing equations, and it is reasonable to conclude that GEP-based modelling has the best performance among all the mentioned scour prediction equations.

Figure 10 compares the GEP-based predicted equilibrium scour depth with four existing equations: (a) Laursen and Toch equation (Laursen & Arthur 1956); (b) Melville and Coleman equation (Melville & Coleman 2000); (c) HEC-18 equation (Arneson et al. 2012); and (d) Sheppard/Melville equation (Sheppard et al. 2014). Larsen and Toch and Melville's equations provide satisfactory results, but HEC-18 and Sheppard's equations overestimated the equilibrium scour depth. The proposed GEP-based modelling renders more accurate predictions of equilibrium scour depth. Its findings are mostly in the 1:1 range. This observation is perfectly depicted in the scatter plots.

Sensitivity analysis has been carried out to identify the impact of the various input variables on the scour depth prediction. This analysis facilitates the selection of the most sensitive variable for the accurate prediction of the scour depth. All the ten input variables used for the GEP formulation are examined, and the results of the sensitivity analysis regarding the validation are illustrated in Table 4. The sensitivity analysis is carried out by eliminating one input variable for each model. The main performance criteria of this process are the determination coefficient (R²). The results presented in Table 4 indicate that the pier spacing has the most significant influence in terms of scour depth prediction in comparison with the other input variables.

For the safety of bridge piers, an accurate estimation of scour depth is of utmost importance, and for this accurate estimation, the influence of various parameters on the scour depth needs to be studied thoroughly. The present study provides a deep understanding of the influencing parameters both experimentally and analytically. In addition, an alternate formulation using the GEP technique is also proposed for reliable scour depth prediction. Efficacy of the proposed GEP formulation is proved by comparing the results with the four well known regression-based equations, namely (a) Laursen and Toch equation; (b) Melville and Coleman equation; (c) HEC-18 equation; and (d) Sheppard/Melville equations. The findings of the present study are as follows:

• Experimental results show that the 1.5D ≤ C_s ≤ 3D scour hole is formed combined for both the piers, and the sediment deposition took place in the form of one tail scour deposition. For the 3.5D ≤ C_s ≤ 4.5D piers behaved partially independent, two separate scour holes are observed upstream of the piers, but the deposition is combined for both the piers. For the C_s = 5D piers, these behaved completely independent, two separate scour holes are observed, and the sediment deposition is observed at the rear of both the piers (two tails of scour deposition). The maximum scour depth is attained for the velocity of flow 0.3 m/s in the case of C_s = 2.5D.

• The prediction capability of the GEP formulation is compared with the well known traditional equations, and the statistical measures of the performance parameters R², MAE, and RMSE were 0.951, 0.00133 and 0.00113, respectively.

• Results of the sensitivity analysis showed that the pier spacing and rate of flow played a sensitive role in the scour depth estimation.

Finally, the findings of the present study concluded that the proposed model is capable of predicting the reliable scour depth around twin bridge piers.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.